Problem: Solve the exponential equation for $x$. 2 8 x − 4 ⋅ 16 3 x − 5 = 16 9 x + 2 2\^{8x-4}\cdot 16\^{ 3x-5}=16\^{ 9x+2} $x=$
Answer: The strategy Let's write $2$ in base $16$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $16$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 2 8 x − 4 ⋅ 16 3 x − 5 = ( 16 1 4 ) 8 x − 4 ⋅ 16 3 x − 5 = 16 2 x − 1 ⋅ 16 3 x − 5 = 16 2 x − 1 + ( 3 x − 5 ) = 16 5 x − 6 ( 2 = 16 1 4 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 2\^{8x-4}\cdot 16\^{ 3x-5}&=(16\^{ \frac 14})\^{8x-4}\cdot 16\^{ 3x-5} &&&&(2=16\^{ \frac14})\\\\ &=16\^{C{2x-1}}\cdot 16\^{ {3x-5}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=16\^{ C{2x-1} \ + \ ({3x-5}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=16\^{ 5x-6} \end{aligned} Solving the linear equation We obtain the following equation. 16 5 x − 6 = 16 9 x + 2 16\^{ 5x-6}=16\^{ 9x+2} Now we can equate the exponents and solve for $x$. $\begin{aligned} 5x-6&=9x+2\\\\ x &= -2\end{aligned}$ The answer The answer is $x=-2$. You can check this answer by substituting $\it{x=-2}$ in the original equation and evaluating both sides.